A sinusoidal phase modulating absolute distance measurement (ADM) interferometer combining frequency-sweeping interferometry (FSI) and multi-wavelength interferometry (MWI) is proposed in this paper. The swept frequency in FSI and the wavelengths for MWI are calibrated by an optical frequency comb, so the distance measurement can be directly traced back to the SI definition of a meter. With a simple optical structure, an ADM interferometer consisting of a measurement interferometer and a monitor interferometer is constructed without polarization optics. A near-infrared external cavity diode laser (ECDL) calibrated by an optical frequency comb is used as a work source of the measurement interferometer for frequency sweeping and hopping. The monitor interferometer using a He-Ne laser runs parallel to the measurement interferometer to monitor the fluctuation of the measured distance during the measurement. Experiments for absolute distance measurements in a range of 8.25 m were carried out to verify the feasibility of the proposed ADM interferometer. The experimental results show that the maximum measurement error is less than 1 μm compared with an incremental-type laser interferometer.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The measurements of distances in the order of several meters to tens of meters with a relative measurement uncertainty of less than 10−6 have been getting increased attention in the application of large-scale metrology. Conventional laser interferometers measure a relative distance by accumulating the incremental movements of the target mirror continuously using the laser wavelength as the ruler, yields a short non-ambiguity of half of the laser wavelength. Without movement of the target mirror, the measurement of absolute distance with an extended non-ambiguity range can be realized by multi-wavelength interferometry (MWI) using one or several synthetic wavelengths generated by two or more accurately known wavelengths. The multiple wavelengths can be employed in parallel based on lasers stabilized to the absorption lines of selected atoms or molecules , synchronized solid-state lasers , or frequency comb referenced multi-wavelength sources [3, 4]. They can also be employed in sequence to simplify the system’s structure by using one external cavity diode laser (ECDL) [5, 6]. High resolution can be achieved by selecting a small synthetic wavelength. However, in order to overcome the ambiguity, a chain of increasing synthetic wavelengths should be generated to cover the measuring distance, and MWI becomes a very complex solution for large ranges.
A direct interferometric measurement of the actual absolute distance without range-limitation can be realized by frequency-sweeping interferometry (FSI) [7–10], which is performed by counting temporal “synthetic fringes” without ambiguity during frequency sweeps. One key problem of the FSI is the exact measurement of the frequency change, which is generally implemented by using stable reference interferometers [8, 9] and Fabry-Perot interferometers [10, 11]. In order to improve the range and the measurement accuracy of the frequency-sweeping, a high-resolution saturation spectroscopy of iodine transitions is used to generate physically stable frequency reference markers . Recently, optical frequency comb (OFC) has revolutionized the fields of spectroscopy and been used as a calibrating optical source in FSI [13, 14], as the comb can provide a few hundreds of thousands of narrow line width wavelengths over a broad optical spectral range [15, 16]. When referenced to the signal of atomic clocks, distance measurements referenced to optical frequency combs can be directly traced back to the SI definition of the meter. A major drawback of FSI is the sensitivity to movement of the target, because a small target movement of the order of one wavelength during a measurement will be interpreted as a movement of one synthetic wavelength. Several methods have been proposed to correct the movement by data analysis  or adding a second laser [4, 18]. Though the measurement accuracy of FSI can only reach micron level due to the limited frequency scanning range of the ECDL without mode-hop, it is sufficient for coarse measurement to obtain a pre-value of the distance. Another way to realized absolute distance measurement (ADM) using frequency-swept technique is the frequency-modulated continuous-wave (FMCW) ladar [19–21], and the swept frequency can also be calibrated by an OFC . As a well-known time-of-flight technique, the instantaneous optical frequency of an ECDL is continuously modulated to a periodic saw-tooth profile in FMCW ladar, and the time-of-flight is measured by detecting the beat frequency between the reference and measurement light beams. However, the measurement uncertainty is greatly affected by the control linearity of frequency modulation and also the free-running frequency change of the laser beam caused by air fluctuation .
High resolution absolute distance measurement (ADM) in large ranges can be realized by combining FSI and MWI, in which FSI is used for coarse measurement and MWI for fine measurement, and the feasibility has been proved in  and  in short distances. A distance of 2 m with a linear error of 0.4 μm compared with a counting interferometer is achieved in , in which a second interferometer is used as a length reference for FSI, and two ECDLs stabilized to two different atom absorption lines are used to generate a synthetic wavelength of 42 μm for MWI. However, the stability of the length reference is influenced by the measurement environment, and the laser frequencies referenced to the atom absorption lines cannot generate arbitrary synthetic wavelengths for the usable atom absorption lines are some fixed and isolated points. In addition, the interference signals in  are processed by conventional homodyne demodulation with two-detector quadrature-detection technique, in which a cyclic error will be introduced by the non-quadrature interference signals, unequal gain of the two detectors and laser power drift. To make the output frequency of an ECDL be modulated to follow a pre-assigned profile which is needed to measure a given length, an OFC is exploited in . Heterodyne phase detection without polarization optics is implemented to avoid the cyclic error that could be caused by the imperfection of polarization optics. However, the movements of the target during the frequency sweeping and hoping cannot be corrected using the heterodyne interferometer proposed in . For a length of ~1.195 m which is made up by multiple reflections installed on a Zerodur plate of low thermal expansion, a measurement uncertainty of 17 nm is achieved. While in practical applications, especially for long distance measurement in sub-optimum environments, the measurement results will be greatly affected without movement correction. To sum up, to realize high accuracy absolute distance measurement in long distances, the key problems in the combination method of FSI and MWI include: (1) the improvement of the interferometric phase measurement accuracy; (2) the correction of the distance fluctuation during frequency sweeping and hoping; (3) the traceability of the distance measurement.
Besides the conventional homodyne demodulation with two-detector quadrature-detection technique mentioned above, conventional heterodyne technique is also widely-used in the FSI and MWI with its strong anti-interference ability [12, 14], while the accuracy is limited by the periodic nonlinear error introduced by the frequency mixing and polarization mixing caused by the non-orthogonal polarized laser beams and the non-perfect polarizing optics, respectively [23, 24]. In this paper, to improve the interferometric phase measurement accuracy, a sinusoidal phase modulating interferometer (SPMI), which has simple optical configuration with no polarization optics, is constructed for phase demodulation in absolute long distance measurement. With high frequency sinusoidal phase modulation, a phase carrier is generated which up-converts the desired phase signal onto the sidebands of the carrier frequency, making the phase demodulation have less sensitivity to the optical and electronic noises [25–28]. And some improved algorithms about sinusoidal phase modulating technique with electro-optic modulator (EOM) have been studied in our previous works [29, 30]. To correct the distance fluctuation during frequency sweeping and hoping, an ADM interferometer consisting of a measurement interferometer and a monitor interferometer with identical optical paths is developed. The laser sources of the two interferometers belong to different spectral bands and they are combined and separated by dichroic mirrors. The swept frequency in FSI and the wavelengths for MWI are calibrated by an OFC which is referenced to atomic clocks, so the absolute distance measurement is directly traceable to the time standard.
2. Sinusoidal phase modulating ADM interferometer
2.1 Optical configuration
As shown in Fig. 1, the sinusoidal phase modulating ADM interferometer is composed of two interferometers with identical optical paths. One is the measurement interferometer using a near-infrared ECDL calibrated by an OFC, and the other is a monitor interferometer using a He-Ne laser. The laser emitting from the ECDL is divided into to two parts by a fiber coupler. One part is used for wavelength monitoring, and the other part is combined with the He-Ne laser by a dichroic mirror (DM1). The combined lasers are introduced to a sinusoidal phase modulating interferometer (SPMI) consisting of P, BS1, EOM, R1 and R2, and the optical paths of the two interferometers are modulated simultaneously by an EOM. Then, the interference beams are separated by another dichroic mirror (DM2). After filtering out the noise induced by the nonideal performance of the dichroic mirror with optical filters (F1 and F2), the interference beams of the measurement and monitor interferometers are detected by two photodetectors (PD1 and PD2), respectively, and the phases of the two interferometers are demodulated in real time (more on this in the next section). In order to evaluate the validity of the ADM interferometer, a commercial incremental-type laser interferometer consisting of M, BS2, R3 and R4 is conducted for relative distance measurement (RDM). The measurement beam of the RDM interferometer is lower than that of the ADM interferometer. Both of the probe retro-reflectors for ADM (R2) and RDM (R4) are mounted on a long translation stage with length of ~8 m and the environmental parameters including temperature (T), pressure (P), humidity (H) and concentrations of CO2 (x) are monitored by sensors distributed along the optical path.
2.2 Phase demodulation
Assuming that the modulating signal applied to the EOM isEq. (2), we can obtain
From Fig. 1 we can see that, single-frequency laser is used in the SPMI which has simple optical configuration with no polarization optics, so the sinusoidal phase modulating technique has no problems about frequency mixing and polarization mixing compared with conventional heterodyne technique. From Eq. (3), the desired phase φj(t) is up-converted onto the sidebands of the carrier frequency which is much more higher than the Doppler frequency caused by the moving target, and the laser intensity has no influence on the demodulated phase according to Eq. (6), so the sinusoidal phase modulating technique has less sensitivity to the low-frequency noises and the laser power drift compared with the conventional homodyne demodulation with two-detector quadrature-detection technique.
3. Absolute distance measurement combining FSI and MWI
Measurement of the absolute distance consists of two steps. Firstly, the frequency of ECDL sweeps from νE0 to νE1 for coarse measurement by FSI, then hops toνE2, …, νEn in sequence for fine measurements by MWI. Combinations of every νEi (i = 1, 2, …, n) andνE0 can produce a set of synthetic wavelengths λSi (νE0, νEi) with λS1 > λS2 > … > λSn. Supposing that Li represents a measurement of the absolute distance L based on the synthetic wavelength λSi, to avoid fringe order ambiguity, the following inequality should be satisfied
In order to improve the stability and measurement accuracy of the laser frequency, the instantaneous frequency of the ECDL νEi (i = 0, 1, …, n) is locked to the OFC, which is referenced to the signal of atomic clocks. During the measurement, a fluctuation of the measured distance induced by the mechanical vibration and deformation is monitored by a monitor interferometer. It is assumed that the frequency of the He-Ne laser is νH, the air refractive indexes corresponding to νH and νEi are nH and nEi, respectively, and the fractional fringe corresponding to νEi is εEi (i = 0, 1, …, n). During the frequency sweeping or hopping of the ECDL from νE0 to νEi (i = 1, …, n), and the variations of the interference fringes with respect to the monitor interferometer and the measurement interferometer are denoted as ∆NHi and ∆NEi, respectively. With the frequency of the ECDL sweeping from νE0 to νE1, the absolute distance can be coarsely determined as
After frequency-sweeping, the frequency of the ECDL hops toνE2 to perform fine distance measurement by using MWI. With the frequency of ECDL changes fromνE0 toνE2, a second order synthetic wavelength can be generated as λS2 = c/(nE2νE2−nE0νE0). By applying the mathematical analysis described in , the integral fringe of the synthetic wavelength λS2 can be accurately determined byEqs. (10)-(13).
4. Experiments and results
4.1 Experimental setup
The experimental setup of the ADM interferometer was constructed as shown in Fig. 2. The laser sources and measurement system of near end is shown in Fig. 2(a), and the measurement system of far end is shown in Fig. 2(b). The laser source of the measurement interferometer is an ECDL (TLB6700, New Focus) which offers a tunable range of 765-781 nm in wavelength with a linewidth of less than 300 kHz. An OFC (FC1500-250, Menlo System) centered at a wavelength of 780 nm with the bandwidth of 20 nm and average power of 180mW is exploited as the optical frequency ruler. The repetition rate (fr) and the carrier-offset frequency (fo) of the OFC are 250 MHz and 20 MHz, respectively. To lock the ECDL to the OFC with a beat note (fb) of 20 MHz, a beat detection unit and a laser locking unit (SYNCRO-LLE, Menlo System) are used for feedback-controlling the current input of the ECDL. Being locked to the kth mode of the OFC, the frequency of the ECDL can be expressed as νE = kfrep + 2fo + fb, where k is determined by a wavelength meter (WSU 30, Highfinesse) with uncertainty of 30 MHz, and frep, fo and fb are locked to a frequency reference (GPS-10, Menlo System) with relative stability of 5 × 10−12 in 1 s averaging time, respectively. The frequency-sweeping is performed by applying a linear voltage to the PZT input of the ECDL using a single-chip controlled DA converter (DAC8562, 16 bit, Texas Instruments). The output voltage of the DA converter is −2.5 V ~ + 2.5 V and the corresponding frequency-sweeping range of ECDL is ~74 GHz (0.15 nm). The output voltage is tuned at a speed of 1 V/s, and the Doppler frequency caused by frequency-sweeping is ~789 Hz at a distance of 8 m. The frequency hopping of the ECDL is realized by varying the external cavity length using a DC motor controlled by a computer. Four synthetic wavelengths with decreasing values are generated by frequency sweeping and hopping in the experiments, and one group of the selected wavelengths and generated synthetic wavelengths is shown in Table 1. λEi = c/νEi (i = 0, 1, 2, 3, 4) is the vacuum wavelength of the ECDL and λSi (i = 1, 2, 3, 4) is the ith order synthetic wavelength.
The laser source of the monitor interferometer is a He-Ne laser (5517D, Agilent), with a wavelength of 632.991 362 nm which is calibrated by the OFC. The sinusoidal phase modulation is introduced by an EOM (EO-PM-NR-C1, Thorlabs) with lithium niobate crystal, and the modulating signal with a frequency of 6 kHz is generated by a function generator (AFG3102, Tektronix) in tandem with a high voltage amplifier (HVA200, Thorlabs) with a voltage gain of β = 20. The carrier signal and the detected signals of PD1 and PD2 are converted into digital signals by a simultaneous 4-chanel data acquisition card (PCI-9812, Adlink) with a sampling rate of 100 kHz. The demodulation of phases corresponding to the measurement interferometer and the monitor interferometer are performed by a computer based on Labview workbench according to the theory described in section 2.2. To evaluate the performance of the ADM interferometer, a commercial incremental-type He-Ne laser interferometer (XL80, Renishaw) is installed along the same measurement optical path of the target retro-reflector travelling on a long granite air-bearing stage of ~8 m for relative distance measurement. The refractive index of air was estimated using Ciddor’s equation with environmental parameters acquired by a precision data acquisition system (1586A Super-DAQ, Fluke).
4.2 Frequency stability of the OFC-locked ECDL
A frequency stability test of 100 minutes was implemented with the ECDL locked to an adjacent mode of the OFC, and the beat frequency was 20 MHz. As shown in Fig. 3(a), the beat frequency is monitored by a Spectrum Analyzer (N9000B CXA, Keysight Technologies), and the signal-to-noise ratio is ~40 dB with the Resolution Bandwidth (RBW) and Video Bandwidth (VBW) of 100 kHz. The frequency of the OFC-locked ECDL can be calculated by νE = kfrep + 2fo + fb, in which the comb mode k determined by the wavelength meter is 1536930. As frep, fo and fb were recorded simultaneously with a gate time of 1s, thus the variation of optical frequency in 100 min with a gate time of 1s could be calculated. As shown in Fig. 3(b), the Allan deviation of the optical frequency is 2.19 × 10−12 in 1s averaging time, and decreased to 2.48 × 10−13 in 10 s averaging time. The relative stability of the used frequency reference is less than 5 × 10−12 in 1 s averaging time, thus the combined relative uncertainty is less than 5.46 × 10−12 at 1 s averaging.
4.3 Distance measurement results
Several experiments were implemented to verify the feasibility of the proposed ADM interferometer and the measurement results were compared with an incremental-type laser interferometer, whose zero data was set to the offset distance of the ADM interferometer. The air-bearing of the granite stage was turned on during movement and turned off during the measurement of absolute distance to avoid air-induced stage vibration. The first experiment was for stability test of the ADM interferometer. A distance of about 4.25 m was measured repeatedly for 20 times, and the distance fluctuation during the measurement was monitored by the RDM interferometer which was reset to zero at the beginning of the experiment. As shown in Fig. 4, the measurement results of ADM and RDM have the same drift trend and the maximum residual without the offset distance (the first read of the ADM) is 0.36 μm with a standard deviation of 0.18 μm. The second experiment was for long distance measurement, which was performed by translating the measurement retroreflector over a distance of 8 m with a step of 0.5 m and an offset distance of 0.25 m. The measurement results are shown in Fig. 5(a), which indicates that the maximum residual without offset is 0.85 μm with a standard deviation of 0.48 μm. The third experiment was for testing the resolution of the ADM interferometer for long distance measurement. The measurement retroreflector was mounted on a linear stage (P-753.1CD, PI) which was placed at an offset distance of ~8.25 m. The ADM and RMD were conducted simultaneously as the linear stage moving 15μm with a step of 3 μm. The measurement results are shown in Fig. 5(b), which indicates that the maximum residual without offset is −0.40 μm with a standard deviation of 0.29 μm. All the experiments confirm that the proposed ADM interferometer can realized submicron accuracy for the range up to 8 m compared with the commercial RDM interferometer.
According to Eq. (9), the measurement result of FSI can be rewritten asEqs. (15) and (16):
It has been proven in section 4.2 that the wavelength uncertainty of the ECDL locked to the OFC is 5.46 × 10−12 at 1 s averaging. As shown in Table 1, the shortest synthetic wavelength used in the experiment is 66.971 μm, so λSi /λE≈86. The contribution of the wavelength uncertainty to the distance measurement uncertainty is 6.64 × 10−10L.
From Eqs. (16) and (18) we can see that, the laser wavelength uncertainty of monitor interferometer, which is <1 × 10−8 for the used He-Ne laser, has negligible influence on the fractional synthetic fringe. So the distance measurements can be performed with comparatively low demands on the stability of the monitor interferometer. The uncertainty of the fractional synthetic fringe is mainly determined by the uncertainty of phase demodulation. Theoretically, the uncertainty of measured phase based on the phase generated carrier demodulation algorithm is less than 0.1°. However, it suffers from several factors such as the high-frequency noise coming from the EOM electronics, the inherent noise and harmonic distortion of the data acquisition card, the mechanical vibration and laser beam fluctuation. All of these factors make the uncertainty of the measured phase increase to 1.5° in the experiment, and the contribution to distance measurement uncertainty corresponding to the shortest synthetic wavelength is 0.28 μm.
Considering all the uncertainty sources, the total measurement uncertainty of the ADM interferometer isEq. (20). The distance measurement of 8 m with step of 0.5 m has been performed with standard deviation of 0.48 μm, which is a little larger than the estimation of measurement uncertainty. It could be caused by the air-bearing switching of the granite stage for the ADM and the RDM.
It should be noted that, for the proposed ADM interferometer, the frequency sweeping and hoping are implemented sequentially by an OFC-calibrated ECDL to accomplish the measurement of FSI and MWI. Compared with the ADM interferometer which implements MWI by generating multiple wavelengths simultaneously with several different lasers and expands non-ambiguity range by constructing another microwave ranging system, the structure of the laser source for the proposed ADM interferometer is greatly simplified and the coarse and fine measurement share the same optical configuration and signal processing system. While the update rate of the measurement is limited, for it needs some time to perform the frequency sweeping and hoping, and the frequency calibration by the OFC. In the experiments, the processes of frequency sweeping and hoping took about 12 s, in which 3 s is used to perform frequency sweeping by the PZT of the ECDL with a sweep rate of 25 GHz/s in a range of ~75 GHz, and 9 s is used to perform frequency hopping by the DC motor of the ECDL with a speed of 1 nm/s in a range of ~9 nm (as shown in Table 1). Plus the time consumed by frequency calibration and data recording, the measurement period is about 30 s. Thus, the proposed ADM interferometer is suitable for the target with low real-time requirements, such as the large-scale metrology, the calibration of large-scale gauge blocks, and so on. The measurement speed can be enhanced by improving the scanning speed of PZT and DC motor of the ECDL, and the bottle neck we need to overcome is to shorten the time of frequency calibration for all the wavelengths in FSI and MWI.
In this paper, a sinusoidal phase modulating ADM interferometer combining FSI and MWI is proposed. A near-infrared ECDL is used as the laser source of the measurement interferometer to perform frequency sweeping for coarse measurement by FSI, and frequency hopping for fine measurement by MWI. The frequency of the ECDL is calibrated by an OFC, and a frequency stability of 5.46 × 10−12 is achieved at 1 s averaging. A monitor interferometer which has an identical optical path with the measurement interferometer is used to compensate the distance fluctuation during the measurement. The minimum generated synthetic wavelength is about 67 μm, and several experiments were implemented for the feasibility verification of the proposed ADM interferometer compared with an incremental-type laser interferometer. Repeated experiments of 20 times at a distance of 4.25 m indicate that a standard deviation of 0.18 μm with a maximum residual of 0.36 μm is realized. The measurement results of 8 m with an increment of 0.5 m show that the maximum residual is 0.85 μm with a standard deviation of 0.48 μm. A step of 3 μm at a distance of 8.25 m was measured in a range of 15 μm, which achieved a standard deviation of 0.29 μm with a maximum residual of −0.40 μm. These measurement results indicate a great potential of the proposed method in the applications of large-scale metrology.
National Natural Science Foundation of China (51527807 and 51475435); Zhejiang Provincial Natural Science Foundation of China (LZ18E050003); Program for Changjiang Scholars and Innovative Research Team in University (IRT_17R98).
Authors acknowledge the financial support from the 521 Talent Project and Science Foundation of Zhejiang Sci-Tech University.
References and links
1. J. E. Decker, J. R. Miles, A. A. Madej, R. F. Siemsen, K. J. Siemsen, S. de Bonth, K. Bustraan, S. Temple, and J. R. Pekelsky, “Increasing the range of unambiguity in step-height measurement with multiple-wavelength interferometry--application to absolute long gauge block measurement,” Appl. Opt. 42(28), 5670–5678 (2003). [CrossRef] [PubMed]
2. S. Azouigui, T. Badr, J. P. Wallerand, M. Himbert, J. Salgado, and P. Juncar, “Transportable distance measurement system based on superheterodyne interferometry using two phase-locked frequency-doubled Nd:YAG lasers,” Rev. Sci. Instrum. 81(5), 053112 (2010). [CrossRef] [PubMed]
3. Y. Salvadé, N. Schuhler, S. Lévêque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt. 47(14), 2715–2720 (2008). [CrossRef] [PubMed]
4. G. Wang, Y. S. Jang, S. Hyun, B. J. Chun, H. J. Kang, S. Yan, S. W. Kim, and Y. J. Kim, “Absolute positioning by multi-wavelength interferometry referenced to the frequency comb of a femtosecond laser,” Opt. Express 23(7), 9121–9129 (2015). [CrossRef] [PubMed]
5. L. Sheng-Hua and L. Cheng-Chung, “Measuring large step heights by variable synthetic wavelength interferometry,” Meas. Sci. Technol. 13(9), 1382–1387 (2002). [CrossRef]
6. J. Jin, Y.-J. Kim, Y. Kim, S.-W. Kim, and C.-S. Kang, “Absolute length calibration of gauge blocks using optical comb of a femtosecond pulse laser,” Opt. Express 14(13), 5968–5974 (2006). [CrossRef] [PubMed]
8. X. Jia, Z. Liu, L. Tao, and Z. Deng, “Frequency-scanning interferometry using a time-varying Kalman filter for dynamic tracking measurements,” Opt. Express 25(21), 25782–25796 (2017). [CrossRef] [PubMed]
9. L. Hartmann, K. Meiners-Hagen, and A. Abou-Zeid, “An absolute distance interferometer with two external cavity diode lasers,” Meas. Sci. Technol. 19(4), 045307 (2008). [CrossRef]
10. J. E. Decker, A. Cabral, G.-S. Peng, and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 58790L (2005). [CrossRef]
11. A. Cabral, “Accuracy of frequency-sweeping interferometry for absolute distance metrology,” Opt. Eng. 46(7), 073602 (2007). [CrossRef]
12. G. Prellinger, K. Meiners-Hagen, and F. Pollinger, “Spectroscopicallyin situtraceable heterodyne frequency-scanning interferometry for distances up to 50 m,” Meas. Sci. Technol. 26(8), 084003 (2015). [CrossRef]
13. S. Hyun, Y.-J. Kim, Y. Kim, J. Jin, and S.-W. Kim, “Absolute length measurement with the frequency comb of a femtosecond laser,” Meas. Sci. Technol. 20(9), 095302 (2009). [CrossRef]
14. X. Wu, H. Wei, H. Zhang, L. Ren, Y. Li, and J. Zhang, “Absolute distance measurement using frequency-sweeping heterodyne interferometer calibrated by an optical frequency comb,” Appl. Opt. 52(10), 2042–2048 (2013). [CrossRef] [PubMed]
16. T. W. Hänsch, “Nobel Lecture: Passion for precision,” Rev. Mod. Phys. 78(4), 1297–1309 (2006). [CrossRef]
18. S. Han, J. D. Ellis, J. Guo, Y. Guo, C. Lu, G. Liu, B. Liu, F. Chen, Z. Zhuang, X. Xu, and Y. Gan, “The correction of vibration in frequency scanning interferometry based absolute distance measurement system for dynamic measurements,” Proc. SPIE 9677, 96772F (2015).
19. R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–38 (2001). [CrossRef]
20. E. Baumann, F. R. Giorgetta, I. Coddington, L. C. Sinclair, K. Knabe, W. C. Swann, and N. R. Newbury, “Comb-calibrated frequency-modulated continuous-wave ladar for absolute distance measurements,” Opt. Lett. 38(12), 2026–2028 (2013). [CrossRef] [PubMed]
22. W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. L. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning, ” CIRP Ann. – Manufa. Techn. 64(2), 773–796 (2015).
23. S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng. 26(4), 448–455 (2002). [CrossRef]
24. P. Hu, Y. Bai, J. Zhao, G. Wu, and J. Tan, “Toward a nonlinearity model for a heterodyne interferometer: not based on double-frequency mixing,” Opt. Express 23(20), 25935–25941 (2015). [CrossRef] [PubMed]
25. T. Suzuki, T. Ohizumi, T. Sekimoto, and O. Sasaki, “Disturbance-free distributed Bragg reflector laser-diode interferometer with a double sinusoidal phase-modulating technique for measurement of absolute distance,” Appl. Opt. 43(23), 4482–4487 (2004). [CrossRef] [PubMed]
26. Y. Kawata, K. Hyashi, and T. Aoto, “Two-wavelength interferometer based on sinusoidal phase modulation with an acetylene stabilized laser and a second harmonic generation,” Opt. Express 23(12), 16024–16034 (2015). [CrossRef] [PubMed]
27. T. Suzuki, T. Okada, O. Sasaki, and T. Maruyama, “Real-time vibration measurement using a feedback type of laser diode interferometer with an optical fiber,” Opt. Eng. 36(9), 2496–2502 (1997). [CrossRef]
28. O. Sasaki, J. Xin, S. Choi, and T. Suzuki, “Profile measurement of thin films by backpropagation of multiple-wavelength optical fields with two sinusoidal phase-modulating interferometers,” Opt. Commun. 356, 578–581 (2015). [CrossRef]
29. S. Zhang, L. Yan, B. Chen, Z. Xu, and J. Xie, “Real-time phase delay compensation of PGC demodulation in sinusoidal phase-modulation interferometer for nanometer displacement measurement,” Opt. Express 25(1), 472–485 (2017). [CrossRef] [PubMed]
30. S. Zhang, B. Chen, L. Yan, and Z. Xu, “Real-time normalization and nonlinearity evaluation methods of the PGC-arctan demodulation in an EOM-based sinusoidal phase modulating interferometer,” Opt. Express 26(2), 605–616 (2018). [CrossRef] [PubMed]
31. F. Pollinger, K. Meiners-Hagen, M. Wedde, and A. Abou-Zeid, “Diode-laser-based high-precision absolute distance interferometer of 20 m range,” Appl. Opt. 48(32), 6188–6194 (2009). [CrossRef] [PubMed]